Dude, again, please read our Visas/Permits guide at
http://raoulschinasaloon.com/index.php?topic=6736.0. It would clarify an awful lot for you...
"Extending a visa" and "providing a visa" are the same thing.
I'm a little confused now...have you changed employers, or are you trying to extend with a current employer? If your Residence Permit was provided by a previous employer, that will change the situation...
A visa, regardless of type, does not make it legal for you to live in China and work for a Chinese company. What you need are a Work Permit and a Residence Permit. A Work (Z) Visa just gets you into the country and gives you 30 days for your school to get the Permits for you. Hopefully, you have a Residence Permit (NOT just a Z Visa) in your passport!
If you already have a Residence Permit, your school might be able to renew or extend it. However, it's very possible that your employer will not be able to extend your Permits if you don't have a degree...it depends on the provincial authorities and your employer's connections. This is something you urgently need to clarify with your school.
If your school can't get you the Work Permit and Residence Permit, it may be that they lack the license needed to legally employ foreigners. It might also be the case that they can't provide the Permits for you specifically, because you don't have a degree.
Do other foreign teachers at your school have Residence Permits issued by that school? If so, it's probably your lack of degree that's the problem. If nobody there has legal Permits from that school, and the school is simply unable to provide them for anyone, then your school is an illegal shop and you really don't want to work there.
For the FEC...yes, your school will need to cancel an FEC from another school before they can get you a new one. However, if your school can get you an FEC, they should also be able to get you proper Work and Residence Permits. Something is starting to smell pretty fishy here...